Problem 63 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|2 x+1| \leq 0$$

Step-by-Step Solution

Answer

x = -\frac{1}{2}

Step 1 - Understand the Absolute Value Inequality

An absolute value inequality \(|2x+1| \leq 0\) represents the distance of \(2x+1\) from 0 is less than or equal to 0. The only way this is true is if \(2x+1 = 0\).

Step 2 - Solve the Equation Inside the Absolute Value

Set the expression inside the absolute value equal to 0: \(2x + 1 = 0\)

Step 3 - Isolate the Variable

Subtract 1 from both sides of the equation to solve for \(x\): \(2x = -1\)

Step 4 - Solve for x

Divide both sides by 2 to isolate \(x\): \(x = -\frac{1}{2}\)

Step 5 - Verify the Solution

Substitute \(x = -\frac{1}{2}\) back into the original inequality to verify. Since \(|2x + 1| \leq 0\) must equal \( |2(-\frac{1}{2}) + 1| = | 0 | = 0\) which is indeed \( \leq 0\). Thus the solution is verified.

Key Concepts

solving inequalities

To solve inequalities, we aim to find the range of values for which the inequality holds true. Here are the basic steps you should follow in general:

• First, identify the inequality sign and understand the inequality's nature (e.g., <, \leq, >, \geq).
• Isolate the variable on one side using algebraic operations like addition, subtraction, multiplication, or division.
• When you multiply or divide both sides of an inequality by a negative number, remember to flip the inequality sign.
• Graph the solution set on a number line if necessary to visually understand the range of values.
• Use the solution to verify if the inequality holds true by substituting values from the solution set back into the original inequality.

For example, the inequality \(|2x+1| \leq 0\) simplifies directly by evaluating the expression inside the absolute value.

absolute value properties

Absolute values represent the distance of a number from zero on a number line, regardless of direction, meaning they are always non-negative. Key properties include:

• \( |a| \geq 0 \) for any real number.
• \( |a| = a \) if \( a \geq 0 \), and \( |a| = -a \) if \( a \< 0 \).
• \( |ab| = |a||b| \) for any real numbers \( a \) and \( b \).
• \( |a+b| \leq |a| + |b| \) (Triangle Inequality).

For the given exercise \(|2x+1| \leq 0\), the absolute value is zero when \(2x+1 = 0\). Absolute values are particularly helpful in inequalities and equations as they simplify comparisons.

linear equations

Linear equations are equations of the first degree, meaning they graph as straight lines and have no exponents higher than 1. To solve them, use steps like:

• Isolate the variable on one side of the equation. This may involve addition, subtraction, multiplication, or division.
• Perform operations equally on both sides to preserve equality.
• Simplify the equation step-by-step until the variable is isolated.
• Always verify your solution by substituting it back into the original equation to make sure both sides are equal.

For instance, in solving \(2x + 1 = 0\):
1. Subtract 1 from both sides: \(2x = -1\).
2. Divide both sides by 2: \(x = -\frac{1}{2}\).
This technique ensures the solution is correctly derived and can be verified easily. Linear equations form the foundation of more complex algebraic concepts and problems.